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Lemma 24.10.1. In the situation above we have

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{A})}( j_!\mathcal{M}, \mathcal{N}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{A}_ U)}( \mathcal{M}, j^*\mathcal{N}) \]

Proof. By the discussion in Modules on Sites, Section 18.19 the functors $j_!$ and $j^*$ on $\mathcal{O}$-modules are adjoint. Thus if we only look at the $\mathcal{O}$-module structures we know that

\[ \mathop{\mathrm{Hom}}\nolimits _{\text{Gr}^{gr}(\textit{Mod}(\mathcal{O}))}( j_!\mathcal{M}, \mathcal{N}) = \mathop{\mathrm{Hom}}\nolimits _{\text{Gr}^{gr}(\textit{Mod}(\mathcal{O}_ U))}( \mathcal{M}, j^*\mathcal{N}) \]

(Recall that $\text{Gr}^{gr}(\textit{Mod}(\mathcal{O}))$ denotes the graded category of graded $\mathcal{O}$-modules.) Then one has to check that these identifications map the $\mathcal{A}$-module maps on the left hand side to the $\mathcal{A}_ U$-module maps on the right hand side. To check this, given $\mathcal{O}_ U$-linear maps $f^ n : \mathcal{M}^ n \to j^*\mathcal{N}^{n + d}$ corresponding to $\mathcal{O}$-linear maps $g^ n : j_!\mathcal{M}^ n \to \mathcal{N}^{n + d}$ it suffices to show that

\[ \xymatrix{ \mathcal{M}^ n \otimes _{\mathcal{O}_ U} \mathcal{A}_ U^ m \ar[r]_{f^ n \otimes 1} \ar[d] & j^*\mathcal{N}^{n + d} \otimes _{\mathcal{O}_ U} \mathcal{A}_ U^ m \ar[d] \\ \mathcal{M}^{n + m} \ar[r]^{f^{n + m}} & j^*\mathcal{N}^{n + m + d} } \]

commutes if and only if

\[ \xymatrix{ j_!\mathcal{M}^ n \otimes _\mathcal {O} \mathcal{A}^ m \ar[r]_{g^ n \otimes 1} \ar[d] & \mathcal{N}^{n + d} \otimes _\mathcal {O} \mathcal{A}_ U^ m \ar[d] \\ j_!\mathcal{M}^{n + m} \ar[r]^{g^{n + m}} & \mathcal{N}^{n + m + d} } \]

commutes. However, we know that

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{M}^ n \otimes _{\mathcal{O}_ U} \mathcal{A}_ U^ m, j^*\mathcal{N}^{n + d + m}) & = \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_!(\mathcal{M}^ n \otimes _{\mathcal{O}_ U} \mathcal{A}_ U^ m), \mathcal{N}^{n + d + m}) \\ & = \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_!\mathcal{M}^ n \otimes _\mathcal {O} \mathcal{A}^ m, \mathcal{N}^{n + d + m}) \end{align*}

by the already used Modules on Sites, Lemma 18.27.9. We omit the verification that shows that the obstruction to the commutativity of the first diagram in the first group maps to the obstruction to the commutativity of the second diagram in the last group. $\square$

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